2017
DOI: 10.5186/aasfmd.2017.161
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Dimension estimates for Kakeya sets defined in an axiomatic setting

Abstract: In this dissertation we define a generalization of Kakeya sets in certain metric spaces. Kakeya sets in Euclidean spaces are sets of zero Lebesgue measure containing a segment of length one in every direction. A famous conjecture, known as Kakeya conjecture, states that the Hausdorff dimension of any Kakeya set should equal the dimension of the space. It was proved only in the plane, whereas in higher dimensions both geometric and arithmetic combinatorial methods were used to obtain partial results.In the firs… Show more

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Cited by 5 publications
(3 citation statements)
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“…Then, by using polar coordinates and Corollary 7.4 in [3], we see that dim H .E/ D dim H .A / D ˛C ˇ. These were the best known bounds prior to this article (see, however, [7,8,13,17] for progress on the corresponding problem in higher dimensions). Note that, because of the min¹ˇ; ˛º term, the bound (1.3) does not distinguish sets in F ˛;˛f rom the (intuitively much larger) sets in F ˛;2˛.…”
Section: Introductionmentioning
confidence: 68%
“…Then, by using polar coordinates and Corollary 7.4 in [3], we see that dim H .E/ D dim H .A / D ˛C ˇ. These were the best known bounds prior to this article (see, however, [7,8,13,17] for progress on the corresponding problem in higher dimensions). Note that, because of the min¹ˇ; ˛º term, the bound (1.3) does not distinguish sets in F ˛;˛f rom the (intuitively much larger) sets in F ˛;2˛.…”
Section: Introductionmentioning
confidence: 68%
“…In [7], Venieri proved that if Ω is d-Alfhors regular then a Ω-Kakeya set X has Hausdorff dimension greater than d+2 2 + 1 2 . Theorem 4 strengthen this result since it gives better estimate for large n and also since we do not make assumption concerning the set of direction Ω.…”
Section: Resultsmentioning
confidence: 99%
“…The Heisenberg Hausdorff dimension of Euclidean Kakeya sets has been studied in [16] where the author showed a lower bound on the dimension of Kakeya sets. In [17], the author studied Kakeya sets for general metric spaces in axiomatic sense.…”
Section: Definition 11mentioning
confidence: 99%